Question

The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices at (0,0), (0,21) and (21,0) is
a)133
b)190
c)233
d)105

Answer 1

Hint: In this question, we have to find out the number of integral points lying inside the triangle given in the question. Therefore, we should first construct the triangle and find out the conditions for an integral point to lie inside that triangle, then, we can count the total number of points satisfying this condition to obtain the required answer.

Complete step-by-step answer:
The triangle given in the question has the vertices at (0,0), (0,21) and (21,0). Therefore, we can construct the triangle as shown in the figure below

Let a point be represented by \[\left( x,y \right)\] where x and y are integers and let this point lie inside the triangle.
We see that as the point \[\left( x,y \right)\] lies inside the triangle, it should lie above the segment AB, therefore its y coordinate should be greater than 0. So, the first condition is
\[y>0\ldots \ldots \ldots \ldots \ldots \ldots \ldots \left( 1.1 \right)\]
Also, for the point \[\left( x,y \right)\] to lie inside the triangle, it should lie to the right of AC, therefore its x coordinate should be greater than 0. So, the second condition is
\[x>0\ldots \ldots .............(1.2)\]
Now, the equation of a straight line with x and y intercept a and b is given by
$\dfrac{x}{a}+\dfrac{y}{b}=1...................(1.3)$
We see that the x and y intercepts of the line to which BC belongs is 21 and 21. Therefore, using the equation (1.3), the equation of BC should be given by
$\dfrac{x}{21}+\dfrac{y}{21}=1\Rightarrow x+y=21...................(1.4)$

Now, for a given value of x, the y coordinate can vary from 1 to the highest integer below the y coordinate of the point on the line corresponding to that value of x. As the y coordinate of the point lying on the line would be given by 21-x from equation (1.4), the points lying in the interior with an x-coordinate will have the y coordinates satisfying
$y<21-x..................(1.5)$
Therefore, the condition for a point lying inside the triangle from equations (1.1), (1.2) and (1.5) is
$1\le x<21\text{ and }1\le y<21-x$

i.e. for x=1, possible y-coordinates are y=1,2 … 19 i.e. 19 integral points
for x=2, y=1,2 ... 18 i.e. 18 integral points
and so on upto
for x=19, y=1 i.e. 1 integral point
for x=20, 0 possible values of y satisfying equation (1.5) and hence 0 integral points
Therefore, considering all the x and y coordinates, the number of integral points lying inside the triangle are
\[19+18+\ldots \ldots +1+0\ldots \ldots \ldots \ldots \ldots ..\left( 1.6 \right)\]
Using the formula that the sum of integers from 1 to n is equal to $\dfrac{n(n+1)}{2}$.
We get the answer of equation 1.6 to be $\dfrac{19\times 20}{2}=190$
Which matches option (b). Hence, option (b) is the correct answer.

Note: We could also have found out the answer by varying y from 1 to 20 and then checking the allowed values of x for a given value of y. The answer, however, would remain the same.